What is the formula for coil galvanometer?
2 Answers
A coil galvanometer is an instrument used to measure small electric currents by deflecting a pointer across a calibrated scale. The operation of a galvanometer is based on the interaction between a magnetic field and a current-carrying coil. To understand its working and derive the formula, we need to consider several aspects of the coil galvanometer.
### Key Components of a Coil Galvanometer
1. **Coil**: Usually a rectangular or circular coil wound around a core.
2. **Magnetic Field**: Created by permanent magnets or electromagnets.
3. **Deflection Angle**: The angle to which the coil rotates in the magnetic field when a current flows through it.
4. **Spring**: Provides a restoring torque to bring the coil back to its initial position.
### Fundamental Concepts
1. **Torque on the Coil**: When a current \( I \) passes through the coil, a torque \( \tau \) is exerted on it, given by the equation:
\[
\tau = n \cdot B \cdot I \cdot A \cdot \sin(\theta)
\]
where:
- \( n \) = number of turns in the coil
- \( B \) = magnetic flux density (in teslas, T)
- \( I \) = current flowing through the coil (in amperes, A)
- \( A \) = area of the coil (in square meters, m²)
- \( \theta \) = angle between the magnetic field and the normal to the coil surface.
2. **Restoring Torque**: The coil is subjected to a restoring torque \( \tau_r \) due to the spring, which opposes the deflection. This torque is proportional to the deflection angle \( \theta \):
\[
\tau_r = k \cdot \theta
\]
where \( k \) is the spring constant, representing the stiffness of the spring.
3. **Equilibrium Condition**: At equilibrium, the torque due to the current equals the restoring torque:
\[
\tau = \tau_r
\]
Therefore, substituting the two equations we have:
\[
n \cdot B \cdot I \cdot A \cdot \sin(\theta) = k \cdot \theta
\]
### Galvanometer Sensitivity and Calibration
- The sensitivity of the galvanometer \( S \) can be defined as the deflection per unit current:
\[
S = \frac{\theta}{I}
\]
Rearranging the previous torque equation and substituting \( \theta \) yields:
\[
I = \frac{n \cdot B \cdot A}{k} \cdot \sin(\theta)
\]
### Full Formula for Coil Galvanometer
If we define the instrument constant \( C \) as:
\[
C = \frac{n \cdot B \cdot A}{k}
\]
The relationship between current and deflection angle can be expressed as:
\[
I = C \cdot \sin(\theta)
\]
For small angles, \( \sin(\theta) \approx \theta \) (in radians), thus simplifying to:
\[
I = C \cdot \theta
\]
### Summary
- **Current Measurement**: The current flowing through the coil galvanometer can be determined from the deflection angle \( \theta \) as follows:
\[
I = C \cdot \theta
\]
- **Calibration**: The constant \( C \) must be determined experimentally and is specific to each galvanometer.
### Conclusion
In practice, galvanometers are used in various applications, including measuring electric current, checking circuit continuity, and other electronic testing scenarios. The design and sensitivity of the coil galvanometer can vary, but the fundamental principles remain consistent across different types of galvanometers.
### Key Components of a Coil Galvanometer
1. **Coil**: Usually a rectangular or circular coil wound around a core.
2. **Magnetic Field**: Created by permanent magnets or electromagnets.
3. **Deflection Angle**: The angle to which the coil rotates in the magnetic field when a current flows through it.
4. **Spring**: Provides a restoring torque to bring the coil back to its initial position.
### Fundamental Concepts
1. **Torque on the Coil**: When a current \( I \) passes through the coil, a torque \( \tau \) is exerted on it, given by the equation:
\[
\tau = n \cdot B \cdot I \cdot A \cdot \sin(\theta)
\]
where:
- \( n \) = number of turns in the coil
- \( B \) = magnetic flux density (in teslas, T)
- \( I \) = current flowing through the coil (in amperes, A)
- \( A \) = area of the coil (in square meters, m²)
- \( \theta \) = angle between the magnetic field and the normal to the coil surface.
2. **Restoring Torque**: The coil is subjected to a restoring torque \( \tau_r \) due to the spring, which opposes the deflection. This torque is proportional to the deflection angle \( \theta \):
\[
\tau_r = k \cdot \theta
\]
where \( k \) is the spring constant, representing the stiffness of the spring.
3. **Equilibrium Condition**: At equilibrium, the torque due to the current equals the restoring torque:
\[
\tau = \tau_r
\]
Therefore, substituting the two equations we have:
\[
n \cdot B \cdot I \cdot A \cdot \sin(\theta) = k \cdot \theta
\]
### Galvanometer Sensitivity and Calibration
- The sensitivity of the galvanometer \( S \) can be defined as the deflection per unit current:
\[
S = \frac{\theta}{I}
\]
Rearranging the previous torque equation and substituting \( \theta \) yields:
\[
I = \frac{n \cdot B \cdot A}{k} \cdot \sin(\theta)
\]
### Full Formula for Coil Galvanometer
If we define the instrument constant \( C \) as:
\[
C = \frac{n \cdot B \cdot A}{k}
\]
The relationship between current and deflection angle can be expressed as:
\[
I = C \cdot \sin(\theta)
\]
For small angles, \( \sin(\theta) \approx \theta \) (in radians), thus simplifying to:
\[
I = C \cdot \theta
\]
### Summary
- **Current Measurement**: The current flowing through the coil galvanometer can be determined from the deflection angle \( \theta \) as follows:
\[
I = C \cdot \theta
\]
- **Calibration**: The constant \( C \) must be determined experimentally and is specific to each galvanometer.
### Conclusion
In practice, galvanometers are used in various applications, including measuring electric current, checking circuit continuity, and other electronic testing scenarios. The design and sensitivity of the coil galvanometer can vary, but the fundamental principles remain consistent across different types of galvanometers.
The formula for a **moving coil galvanometer** depends on the torque experienced by the coil in a magnetic field. The galvanometer is designed to detect small electric currents, and the deflection of the coil is proportional to the current passing through it. The two main formulas are related to the **deflection** and the **sensitivity** of the galvanometer.
### 1. **Torque on the coil**:
The torque \(\tau\) acting on the coil is given by:
\[
\tau = n B A I
\]
Where:
- \(n\) = number of turns of the coil,
- \(B\) = magnetic field strength (in Tesla),
- \(A\) = area of the coil (in square meters),
- \(I\) = current passing through the coil (in Amperes).
### 2. **Deflection of the needle**:
The coil is attached to a spring that provides a restoring torque. At equilibrium, the deflection \(\theta\) of the needle is proportional to the current \(I\):
\[
\theta = \frac{n B A I}{k}
\]
Where:
- \(k\) = restoring constant (torque per unit angle of deflection) of the spring.
This equation shows that the deflection \(\theta\) is proportional to the current \(I\), which is the basic principle of operation for a moving coil galvanometer.
### 3. **Current sensitivity (S)**:
The current sensitivity of a galvanometer is the deflection per unit current, given by:
\[
S = \frac{\theta}{I} = \frac{n B A}{k}
\]
Where:
- \(S\) is the current sensitivity in radians per Ampere.
### 4. **Voltage sensitivity (V_s)**:
Voltage sensitivity, which is the deflection per unit voltage, is related to the resistance \(R\) of the galvanometer:
\[
V_s = \frac{S}{R} = \frac{n B A}{k R}
\]
### Summary:
- **Torque**: \(\tau = n B A I\)
- **Deflection**: \(\theta = \frac{n B A I}{k}\)
- **Current Sensitivity**: \(S = \frac{n B A}{k}\)
- **Voltage Sensitivity**: \(V_s = \frac{n B A}{k R}\)
These formulas describe how the galvanometer works and help in designing instruments with desired sensitivities.
### 1. **Torque on the coil**:
The torque \(\tau\) acting on the coil is given by:
\[
\tau = n B A I
\]
Where:
- \(n\) = number of turns of the coil,
- \(B\) = magnetic field strength (in Tesla),
- \(A\) = area of the coil (in square meters),
- \(I\) = current passing through the coil (in Amperes).
### 2. **Deflection of the needle**:
The coil is attached to a spring that provides a restoring torque. At equilibrium, the deflection \(\theta\) of the needle is proportional to the current \(I\):
\[
\theta = \frac{n B A I}{k}
\]
Where:
- \(k\) = restoring constant (torque per unit angle of deflection) of the spring.
This equation shows that the deflection \(\theta\) is proportional to the current \(I\), which is the basic principle of operation for a moving coil galvanometer.
### 3. **Current sensitivity (S)**:
The current sensitivity of a galvanometer is the deflection per unit current, given by:
\[
S = \frac{\theta}{I} = \frac{n B A}{k}
\]
Where:
- \(S\) is the current sensitivity in radians per Ampere.
### 4. **Voltage sensitivity (V_s)**:
Voltage sensitivity, which is the deflection per unit voltage, is related to the resistance \(R\) of the galvanometer:
\[
V_s = \frac{S}{R} = \frac{n B A}{k R}
\]
### Summary:
- **Torque**: \(\tau = n B A I\)
- **Deflection**: \(\theta = \frac{n B A I}{k}\)
- **Current Sensitivity**: \(S = \frac{n B A}{k}\)
- **Voltage Sensitivity**: \(V_s = \frac{n B A}{k R}\)
These formulas describe how the galvanometer works and help in designing instruments with desired sensitivities.